Stochastic bounds on a single loss: unknown marginals

216 H. Cossette et al. Insurance: Mathematics and Economics 26 2000 213–222 Table 1 Extremal distributions in 3.1, two moments known, infinite spectrum Value of s M µ,σ s W µ,σ s − M µ,σ s 0 s µ σ 2 s − µ 2 + σ 2 µ s δ 2 µ s − µs µs s δ 2 µ s − µ 2 s − µ 2 + σ 2 σ 2 s − µ 2 + σ 2 positively orthant dependent POD, in short. For more details about POD, the interested reader is referred e.g. to Szekli 1995, pp. 144–145.

3. Stochastic bounds on a single loss: unknown marginals

We now examine the construction of bounds on F X when only the support and the first few moments of the marginals F X j , j = 1, 2, . . . , m, are known. We first derive the best stochastic upper and lower bounds on the X j i ’s. For that purpose, we use the following results of Kaas and Goovaerts 1985. Consider a non-negative random variable Y for which only the mean µ, and the standard deviation σ are known. Then, there exists two cumulative distribution functions, M µ,σ and W µ,σ , say, such that M µ,σ s ≤ F Y s ≤ W µ,σ s 3.1 is verified for all s ≥ 0. Explicit expressions for the extremal distributions in 3.1 are provided in Table 1 Table 2 in Kaas and Goovaerts, 1985, where δ 2 stands for the second moment of Y , i.e. δ 2 = EY 2 . When it is further known that there exists an upper bound b for Y , i.e. P [Y ≤ b] = 1, the extremal distributions in 3.1 can be refined as M µ,σ,b s ≤ F Y s ≤ W µ,σ,b s, 3.2 which is verified for all s ≥ 0. Explicit expressions for these extremal distributions are provided in Table 2 Table 1 in Kaas and Goovaerts, 1985. When the skewness γ of Y is also known, tighter bounds M µ,σ,γ and W µ,σ,γ , say, can be found such that M µ,σ,γ s ≤ F Y s ≤ W µ,σ,γ s 3.3 is verified for all s ≥ 0. Explicit expressions for the extremal distributions in 3.3 are provided in Table 3 Table 4 in Kaas and Goovaerts, 1985, where the following symbols are used: δ 3 stands for third moment of Y , i.e. δ 3 = EY 3 , β 1 s = γ + 3σ 2 + µ 3 − sδ 2 δ 2 − sµ , β 2 s = δ 2 − µs µ − s , and α ± = δ 3 − µδ 2 ± q δ 3 − µδ 2 2 − 4σ 2 µδ 3 − δ 2 2 2σ 2 . Table 2 Extremal distributions in 3.2, two moments known, finite spectrum Value of s M µ,σ,b s W µ,σ,b s − M µ,σ,b s 0 s µ − σ 2 b − µ σ 2 s − µ 2 + σ 2 µ − σ 2 b − µ s δ 2 µ σ 2 + µ − bµ − ssb σ 2 + µµ − bss − b s δ 2 µ s − µ 2 s − µ 2 + σ 2 σ 2 s − µ 2 + σ 2 H. Cossette et al. Insurance: Mathematics and Economics 26 2000 213–222 217 Table 3 Extremal distributions in 3.3, three moments known, infinite spectrum Value of s M µ,σ,γ s W µ,σ,γ s − M µ,σ,γ s 0 s α − µ − β 2 ss − β 2 s α − s δ 2 µ σ 2 + µ − sµ − β 1 ssβ 1 s σ 2 + µ − β 1 sµss − β 1 s δ 2 µ s α + µ − sβ 2 s − s µ − β 2 ss − β 2 s s α + σ 2 + µ − sµ − β 1 ssβ 1 s + σ 2 + µ − sµβ 1 s − sβ 1 s σ 2 + µ − β 1 sµss − β 1 s As in 3.2, when it is further known that there exists an upper bound b for Y , the extremal distributions in 3.3 can be refined as M µ,σ,γ ,b s ≤ F Y s ≤ W µ,σ,γ ,b s, 3.4 which is verified for all s ≥ 0. Explicit expressions for these extremal distributions are provided in Table 4 Table 3 in Kaas and Goovaerts, 1985, where the following symbols are used: ζ = γ + 3σ 2 + µ 3 − bδ 2 δ 2 − bµ , β ∗ 2 s = γ + 3σ 2 + µ 3 − b + sδ 2 + bsµ δ 2 − b + sµ + bs . Now, let µ j , σ j and γ j be the mean, the standard deviation and the skewness corresponding to F X j , j = 1, 2, . . . , m. With Tables 1 and 3, we can find the best upper and lower bounds on F X j , j = 1, 2, . . . , m, namely M j s ≤ F X j s ≤ W j s for j = 1, 2, . . . , m, 3.5 where M j resp. W j stands for either M µ j ,σ j or M µ j ,σ j ,γ j resp. W µ j ,σ j or W µ j ,σ j ,γ j . When there exist b j such that F X j b j = 1, j = 1, 2, . . . , m, then 3.5 becomes M b j j s ≤ F X j s ≤ W b j j s for j = 1, . . . , m, 3.6 where M b j j resp. W b j j stands for either M µ j ,σ j ,b j or M µ j ,σ j ,γ j ,b j resp. W µ j ,σ j ,b j resp. W µ j ,σ j ,γ j ,b j , j = 1, 2, . . . , m. Then, ˜ F min s ≤ F X s ≤ ˜ F max s for all s ≥ 0, 3.7 with, for s ∈ R, ˜ F min s = sup x 1 ,x 2 ,... ,x m ∈ Ps max    m X j =1 lim n →∞ M j x j − 1 n − m − 1, 0    , Table 4 Extremal distributions in 3.4, three moments known, finite spectrum Value of s M µ,σ,γ ,b s W µ,σ,γ ,b s − M µ,σ,γ ,b s 0 s α − σ 2 + µ − β ∗ 2 sµ − bs − β ∗ 2 ss − b α − s ζ σ 2 + µ − sµ − β 1 ssβ 1 s σ 2 + µ − β 1 sµss − β 1 s ζ s α + σ 2 + µ − sµ − bβ ∗ 2 s − sβ ∗ 2 s − b σ 2 + µ − β ∗ 2 sµ − bs − β ∗ 2 ss − b s α + σ 2 + µ − sµ − β 1 ssβ 1 s + σ 2 + µ − sµβ 1 s − sβ 1 s σ 2 + µ − β 1 sµss − β 1 s 218 H. Cossette et al. Insurance: Mathematics and Economics 26 2000 213–222 and ˜ F max s = inf x 1 ,x 2 ,... ,x m ∈ Ps min    m X j =1 W j x j , 1    . Of course, when there exist b j such that F X j b j = 1, j = 1, . . . , m, M j and W j in the expressions of ˜ F min and ˜ F max are replaced by M b j j and W b j j , respectively. If the X 1 i , X 2 i , . . . , X m i ’s are POD, then 3.7 is valid with the following improved bounds: ˜ F min s = sup x 1 ,x 2 ,... ,x m ∈ Ps m Y j =1 M j x j and ˜ F max s = 1 − sup x 1 ,x 2 ,... ,x m ∈ Ps m Y j =1 1 − W j x j .

4. Stochastic bounds on the total amount of loss